Theme Essential Question: How are transformations a visual representation of an object moving in our three-dimensional world?
Essential Questions: How are coordinates used to describe the results of translations, reflections, and rotations?
Standards
8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Objectives
The student will apply translational moves to pre-images and describe these moves by adjusting the shift in the x- and y- coordinate. This will include using translational notation.
The student will apply reflectional moves to pre-images and describe these moves by adjusting the shift in the x- and y- coordinate. This will include reflections over x- axis, y-axis, the line y = x, and the line y = -x.
The student will apply rotational moves to pre-images and describe these moves by adjusting the shift in the x- and y- coordinate. This will include rotation about a point other than the origin.
Reflections and/or Comments from your PCSSD 8th Grade Curriculum Team Students should be provided the opportunity to explore and investigate the transformations and relationships using such items as rules and protractors, patty paper, TI84 geometry app (Cabri Jr.), and/or software such as GeoGebra or Geometer’s Sketchpad.
(Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 –page 17) Transformational geometry is about the effects of rigid motions, rotations, reflections and translations on figures. Initial work should be presented in such a way that students understand the concept of each type of transformation and the effects that each transformation has on an object before working within the coordinate system. For example, when reflecting over a line, each vertex is the same distance from the line as its corresponding vertex. This is easier to visualize when not using regular figures. Time should be allowed for students to cut out and trace the figures for each step in a series of transformations. Discussion should include the description of the relationship between the original figure and its image(s) in regards to their corresponding parts (length of sides and measure of angles) and the descriptionof the movement, including the atributes of transformations (line of symmetry, distance to be moved, center of rotation, angle of rotationand the amount of dilation).The case of distance – preserving transformation leads to the idea of congruence It is these distance-preserving transformations that lead to the idea of congruence. Work in the coordinate plane should involve the movement of various polygons by addition, subtraction and multiplied changes of the coordinates. For example, add 3 to x, subtract 4 from y, combinations of changes to x and y, multiply coordinates by 2 then by 12. Students should observe and discuss such questions as ‘What happens to the polygon?’ and ‘What does making the change to all vertices do?’. Understandings should include generalizations about the changes that maintain size or maintain shape, as well as the changes that create distortions of the polygon (dilations). Example dilations should be analyzed by students to discover the movement from the origin and the subsequent change of edge lengths of the figures. Students should be asked to describe the transformations required to go from an original figure to a transformed figure (image). Provide opportunities for students to discuss the procedure used, whether different procedures can obtain the same results, and if there is a more efficient procedure to obtain the same results. Students need to learn to describe transformations with both words and numbers. Through understanding symmetry and congruence, conclusions can be made about the relationships of line segments and angles with figures. Students should relate rigid motions to the concept of symmetry and to use them to prove congruence or similarity of two figures. Problem situations should require students to use this knowledge to solve for missing measures or to prove relationships. It is an expectation to be able to describe rigid motions with coordinates.
Assessment Product Mathematics behind the Art of MC EscherExplore the transformations using principles of MC Escher from this website, and have students work to create a similar art project with examples of each type of transformation included.This will be an ongoing project over several weeks. Recommended schedule for project development:
Week 1: Students are to explore the principles of MC Escher.
Week 2: Students are to study the development of design and prepare their initial sketch.
Week 3: Students are finalize their design work.
Week 4: Museum Walk.
Key Questions
How are the coordinates of a pre-image affected by a translation, reflection, and rotation?
Observable Student Behaviors
The student can perform the transformation and describe the effects of a translation, reflection, and rotation.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary Transformation, pre-image, image, translation, reflection, line of reflections, and rotation
Suggested Activities [see Legend to highlight MCO and HYS]
Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 4.1 , p. 91-94
ABC Mastering the Common Core in Mathematics 9.3-9.5, p.135-141
Gizmo Lessons
8.G.3
Reflections
Reshape and resize a figure and examine how its reflection changes in response. Move the line of reflection and explore how the reflection is translated.
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation.
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
p. 210
JBHM 8th GP3 p. 63-70, 73-78, 83-103 Glencoe Pre-Algebra- p. 506-511 Glencoe Algebra 1- p. 197-203
Highly Recommended:
Nothing available at this time (G.3)The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
This Smart Board activity explores the concepts of line symmetry and rotational symmetry. The closure of the lesson has students create their own logo for a fictitious company.
8.G. 3 Rotation
The objectives are to identify and perform rotations in the coordinate plane.
8.G. 3 Translation
The objectives are to translate figures in the coordinate plane and to describe the translation.
Content:Translations, Reflections, and Rotations
Dates: 3/4-3/8
Theme Essential Question:
How are transformations a visual representation of an object moving in our three-dimensional world?
Essential Questions:
How are coordinates used to describe the results of translations, reflections, and rotations?
Standards
Objectives
Reflections and/or Comments from your PCSSD 8th Grade Curriculum Team
Students should be provided the opportunity to explore and investigate the transformations and relationships using such items as rules and protractors, patty paper, TI84 geometry app (Cabri Jr.), and/or software such as GeoGebra or Geometer’s Sketchpad.
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials.
http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf
(Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 –page 17)
Transformational geometry is about the effects of rigid motions, rotations, reflections and translations on figures. Initial work should be presented in such a way that students understand the concept of each type of transformation and the effects that each transformation has on an object before working within the coordinate system. For example, when reflecting over a line, each vertex is the same distance from the line as its corresponding vertex. This is easier to visualize when not using regular figures. Time should be allowed for students to cut out and trace the figures for each step in a series of transformations. Discussion should include the description of the relationship between the original figure and its image(s) in regards to their corresponding parts (length of sides and measure of angles) and the descriptionof the movement, including the atributes of transformations (line of symmetry, distance to be moved, center of rotation, angle of rotationand the amount of dilation).The case of distance – preserving transformation leads to the idea of congruence
It is these distance-preserving transformations that lead to the idea of congruence.
Work in the coordinate plane should involve the movement of various polygons by addition, subtraction and multiplied changes of the coordinates. For example, add 3 to x, subtract 4 from y, combinations of changes to x and y, multiply coordinates by 2 then by 12. Students should observe and discuss such questions as ‘What happens to the polygon?’ and ‘What does making the change to all vertices do?’. Understandings should include generalizations about the changes that maintain size or maintain shape, as well as the changes that create distortions of the polygon (dilations). Example dilations should be analyzed by students to discover the movement from the origin and the subsequent change of edge lengths of the figures. Students should be asked to describe the transformations required to go from an original figure to a transformed figure (image). Provide opportunities for students to discuss the procedure used, whether different procedures can obtain the same results, and if there is a more efficient procedure to obtain the same results. Students need to learn to describe transformations with both words and numbers.
Through understanding symmetry and congruence, conclusions can be made about the relationships of line segments and angles with figures. Students should relate rigid motions to the concept of symmetry and to use them to prove congruence or similarity of two figures. Problem situations should require students to use this knowledge to solve for missing measures or to prove relationships. It is an expectation to be able to describe rigid motions with coordinates.
Assessment
Product
Mathematics behind the Art of MC Escher Explore the transformations using principles of MC Escher from this website, and have students work to create a similar art project with examples of each type of transformation included.This will be an ongoing project over several weeks. Recommended schedule for project development:
Key Questions
Observable Student Behaviors
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary
Transformation, pre-image, image, translation, reflection, line of reflections, and rotation
Suggested Activities [see Legend to highlight MCO and HYS]
- Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 4.1 , p. 91-94
- ABC Mastering the Common Core in Mathematics 9.3-9.5, p.135-141
Gizmo Lessons- 8.G.3
- Reflections
- Reshape and resize a figure and examine how its reflection changes in response. Move the line of reflection and explore how the reflection is translated.
- Translations
- Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation.
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla- p. 210
JBHM 8th GP3 p. 63-70, 73-78, 83-103Glencoe Pre-Algebra- p. 506-511
Glencoe Algebra 1- p. 197-203
- Highly Recommended:
Nothing available at this time (G.3)The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.Diverse Learners
Homework
Suggested:
Terminology for Teachers
Ethnicity/Culture | Immigration/Migration | Intercultural Competence | Socialization | Racism/Discrimination
High Yield Strategies
Similarities/Differences | Summarizing/Notetaking | Reinforcing/Recognition | Homework/Practice |
Non-Linguistic representation | Cooperative Learning | Objectives/Feedback |
Generating-Testing Hypothesis | Cues, Questions, Organizers
Resources
Professional Texts
Literary Texts
Informational Texts
- See New York Common Core Aligned Task (other resources)
http://schools.nyc.gov/Academics/CommonCoreLibrary/SeeStudentWork/default.htmArt, Music, and Media
Manipulatives
Games
Videos
SMART Board Lessons, Promethean Lessons
- Smartboard Resource Website Smartboard lesson search engine
- 8.G. 3 Symmetry
This Smart Board activity explores the concepts of line symmetry and rotational symmetry. The closure of the lesson has students create their own logo for a fictitious company.- 8.G. 3 Rotation
The objectives are to identify and perform rotations in the coordinate plane.- 8.G. 3 Translation
The objectives are to translate figures in the coordinate plane and to describe the translation.Websites
Other Activities, etc.
Language
Arts
Week 1
Week 2
Week 3
Week 4
Matrix
Week 1
Week 2
Week 3
Week 4
Home K-2
Home 3-6
Home 6-8
Unit 1
Unit 2
Unit 3
Unit 4